The 7-simplex is by definition a seven-dimensional object, so the phrase "can be positioned in 8-space" needs some explanation, or a pointer. I believe that after constructing all the permutations, one discards one coordinate. To center the result on the origin, one could then subtract the mean from all points. Vanzandtj (talk) —Preceding undated comment added 17:06, 23 November 2016 (UTC)
Sure, its a hyperplane on the subspace x1+x2+...+x7=1, or normal vector (1,1,1,1,1,1,1,1).
Here's the 3D version with a triangle (2-simplex). Tom Ruen (talk) 18:17, 23 November 2016 (UTC)